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Theorem ssuzfz 37659
Description: A finite subset of the upper integers is a subset of a finite set of sequential integers. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypotheses
Ref Expression
ssuzfz.1  |-  Z  =  ( ZZ>= `  M )
ssuzfz.2  |-  ( ph  ->  A  C_  Z )
ssuzfz.3  |-  ( ph  ->  A  e.  Fin )
Assertion
Ref Expression
ssuzfz  |-  ( ph  ->  A  C_  ( M ... sup ( A ,  RR ,  <  ) ) )

Proof of Theorem ssuzfz
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 ssuzfz.2 . . . . . . . . . 10  |-  ( ph  ->  A  C_  Z )
21sselda 3418 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  k  e.  Z )
3 ssuzfz.1 . . . . . . . . 9  |-  Z  =  ( ZZ>= `  M )
42, 3syl6eleq 2559 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  k  e.  ( ZZ>= `  M )
)
5 eluzel2 11187 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
64, 5syl 17 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  M  e.  ZZ )
7 uzssz 11202 . . . . . . . . . . . 12  |-  ( ZZ>= `  M )  C_  ZZ
83, 7eqsstri 3448 . . . . . . . . . . 11  |-  Z  C_  ZZ
98a1i 11 . . . . . . . . . 10  |-  ( ph  ->  Z  C_  ZZ )
101, 9sstrd 3428 . . . . . . . . 9  |-  ( ph  ->  A  C_  ZZ )
1110adantr 472 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  A  C_  ZZ )
12 ne0i 3728 . . . . . . . . . 10  |-  ( k  e.  A  ->  A  =/=  (/) )
1312adantl 473 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  A  =/=  (/) )
14 ssuzfz.3 . . . . . . . . . 10  |-  ( ph  ->  A  e.  Fin )
1514adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  A  e.  Fin )
16 suprfinzcl 11073 . . . . . . . . 9  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  sup ( A ,  RR ,  <  )  e.  A )
1711, 13, 15, 16syl3anc 1292 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  A )
1811, 17sseldd 3419 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  ZZ )
1910sselda 3418 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  k  e.  ZZ )
206, 18, 193jca 1210 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( M  e.  ZZ  /\  sup ( A ,  RR ,  <  )  e.  ZZ  /\  k  e.  ZZ )
)
21 eluzle 11195 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  M  <_  k )
224, 21syl 17 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  M  <_  k )
23 zssre 10968 . . . . . . . . . 10  |-  ZZ  C_  RR
2423a1i 11 . . . . . . . . 9  |-  ( ph  ->  ZZ  C_  RR )
2510, 24sstrd 3428 . . . . . . . 8  |-  ( ph  ->  A  C_  RR )
2625adantr 472 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  A  C_  RR )
27 simpr 468 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  k  e.  A )
28 eqidd 2472 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  sup ( A ,  RR ,  <  )  =  sup ( A ,  RR ,  <  ) )
2926, 15, 27, 28supfirege 10620 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  k  <_  sup ( A ,  RR ,  <  ) )
3020, 22, 29jca32 544 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
( M  e.  ZZ  /\ 
sup ( A ,  RR ,  <  )  e.  ZZ  /\  k  e.  ZZ )  /\  ( M  <_  k  /\  k  <_  sup ( A ,  RR ,  <  ) ) ) )
31 elfz2 11817 . . . . 5  |-  ( k  e.  ( M ... sup ( A ,  RR ,  <  ) )  <->  ( ( M  e.  ZZ  /\  sup ( A ,  RR ,  <  )  e.  ZZ  /\  k  e.  ZZ )  /\  ( M  <_  k  /\  k  <_  sup ( A ,  RR ,  <  ) ) ) )
3230, 31sylibr 217 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  k  e.  ( M ... sup ( A ,  RR ,  <  ) ) )
3332ex 441 . . 3  |-  ( ph  ->  ( k  e.  A  ->  k  e.  ( M ... sup ( A ,  RR ,  <  ) ) ) )
3433ralrimiv 2808 . 2  |-  ( ph  ->  A. k  e.  A  k  e.  ( M ... sup ( A ,  RR ,  <  ) ) )
35 dfss3 3408 . 2  |-  ( A 
C_  ( M ... sup ( A ,  RR ,  <  ) )  <->  A. k  e.  A  k  e.  ( M ... sup ( A ,  RR ,  <  ) ) )
3634, 35sylibr 217 1  |-  ( ph  ->  A  C_  ( M ... sup ( A ,  RR ,  <  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756    C_ wss 3390   (/)c0 3722   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Fincfn 7587   supcsup 7972   RRcr 9556    < clt 9693    <_ cle 9694   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811
This theorem is referenced by:  sge0isum  38383
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